Step by step
The question
Question: decide if f(x) = 2x + 5 is continuous at x = 1.
Continuous at x = 1 means three things are true.
One: f(1) has a value.
Two: the limit there exists.
Three: they are equal.
1 Does f have a value?
Put 1 in place of x.
We get 7.
Careful!
Having a value is not enough.
The graph must also arrive at that height from both sides.
Check 1 is true: f(1) is 7.
2 Does the limit exist?
A polynomial has no gaps and no jumps.
So its limit at 1 is just its value: 7.
So the limit is 7.
Check 2 is true: the limit exists.
3 Are they equal?
The limit is 7, and f(1) is 7.
They are equal.
Check 3 is true.
All three checks are true.
So f is continuous at x = 1.
4 Check
Let's check on the graph.
The graph is one piece with no gaps.
It passes through (1, 7) without lifting the pen.
Right!
Let's look back.
First: does f have a value?.
Then: does the limit exist?.
Then: are they equal?.
Last: check.